on causal stochastic equations for log-stable...

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EPJ manuscript No.(will be inserted by the editor)

On causal stochastic equations for log-stable multiplicativecascades

F. G. Schmitt1 and P. Chainais2

1 CNRS FRE 2816 ELICO, Wimereux Marine Station, Universite des Sciences et Technologies de Lille - Lille 1, 28 av. Foch,62930 Wimereux, Francee-mail: [email protected]

2 LIMOS UMR CNRS 6158-Universite Blaise Pascal 63173 Aubiere, France.e-mail: [email protected], http://www.isima.fr/∼chainais

Received: date / Revised version: date Version September 10, 2007

Abstract. We reformulate various versions of infinitely divisible cascades proposed in the literature usingstochastic equations. This approach sheds a new light on the differences and common points of several for-mulations that have been recently provided by several teams. In particular, we focus on the simplificationoccurring when the infinitely divisible noise at the heart of such a model is stable: an independently scat-tered random measure becomes a stable stochastic integral. In a last section we discuss the D-dimensionalgeneralization.

PACS. 02.50.Ey Stochastic processes – 47.27.E- Turbulence simulation and modelling – 5.45.Df Fractals

1 Introduction

Multifractals have been introduced more than two decadesago in the fields of turbulence, geophysics and chaos the-ory. Since then, the multifractal framework has been usedwidely in many fields including turbulence [1,2], precipi-tations [3,4], oceanography [5,6], biology [7–9], chemistry[10] astrophysics [11], finance [12–14], etc. The main prop-erties of multifractal models or data are their high variabil-ity on a wide range of spatial or temporal scales, associatedto scaling intermittent fluctuations and long-range power-law correlations. For a long time, research in this frame-work was mainly devoted to data analysis, or to modeldevelopment using discrete scale constructions. Only re-cently, research has developed on the definition and prop-erties of multifractal stochastic processes. In this frame-work, various recent works [15–20] have presented similarobjects in different ways, involving multifractal stochasticprocesses depending on a continuous parameter. These ob-jects belong to the family of infinitely divisible cascades,a large class of multifractal scalar fields. We propose toreview these presentations of infinitely divisible cascades,and to show how they simplify for a specific family ofstochastic processes: stable processes (including Gaussianprocesses). In this approach, we will use descriptions basedon the framework of multiplicative cascades as well asthe framework of stochastic integrals as proposed for theGaussian case in 1 dimension in [17].

First, we consider cascades in 1 dimension only. Werecall the main definitions and properties of infinitely di-visible (ID) cascades. We also recall the definition using

a stochastic integral proposed in [17] in the special Nor-mal case. We extend the result of [17] describing explicitlyhow the ID framework simplifies for Levy stable randomvariables in 1 dimension. Then we show that most of ourarguments are easier to understand in 1 dimension butgeneralize rather naturally to D ≥ 2 dimensions. More-over, a nice formulation based on the use of a stochas-tic integral with respect to some α-stable noise is given.Finally, we propose to consider separately D spatial di-mensions and 1 time dimension within a unique D + 1dimensional cascade so that causal animated scalar fieldscan be considered. Again a formulation using a stochasticintegral with respect to some α-stable noise is proposedthat makes explicit the main properties of the resultingfield: multifractality both in time and space, homogene-ity, stationarity and causality.

2 From cascades to stochastic integrals: the1D causal case

Examples of realizations of 1D, 2D and 3D infinitely divis-ible cascades are shown on figure 1. Let us first recall thedefinition of a scale invariant infinitely divisible cascade.

2.1 Definitions

Let G(X) be an infinitely divisible distribution with mo-ment generating function G(q) that can be written in thefollowing form G(q) = IE(eqX) = e−ρ(q).

2 F. G. Schmitt, P. Chainais: On causal stochastic equations for log-stable multiplicative cascades

1D 2D 3D

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

t

Qr(t

)

Fig. 1. Examples of realizations of infinitely divisible cascades in 1, 2 and 3 dimensions respectively.

In full generality, let dm(t, r) = g(r)dtdr a positivemeasure on the time-scale half-plane P+ := R × R+. Wewill see below that scale invariance imposes the specialchoice dm(t, r) = 1

r2 dtdr. Let M denote an infinitely divis-ible (ID), additive independently scattered random mea-sure (called an “ID random measure” in the following)distributed by G, and supported on the time-scale half-plane P+ and associated to its so-called control measuredm(t, r). For any subset E of P+, the random measure Mis such that

IE[exp [qM(E)]] = exp [−ρ(q)m(E)] . (1)

For all disjoints subsets E1 and E2, M(E1) and M(E2) areindependent random variables and M(E1∪E2) = M(E1)+M(E2). For more information on random measures, see[21–23], and for a first use for cascades see [15].

Definition 1For a given resolution 0 < ` ≤ 1, let C`(t) the cone ofinfluence defined below for every t ∈ R. An Infinitely Di-visible Cascading measure density (IDC measure density)is a family of processes Q`(t) parametrized by ` of the form

Q`(t) =exp [M(C`(t))]

IE[expM(C`(t))]. (2)

Possible choices for distribution G are the Normal dis-tribution, Poisson distribution, compound Poisson distri-butions, Gamma laws, Levy-stable laws, and infinitely ma-ny others, so that a large variety of choices is available formodelling and applications (see [24]).

Different definitions of the cone C`(t) have been re-cently proposed in the literature. To get causal definitions,we adapt the expressions given in [16,18,19], and consideronly integration on positions (t′, r′) for which t′ ≤ t. The

corresponding choices are 1:CBM

` (t) = {(t′, r′) : ` ≤ r′ ≤ 1, t− r′ ≤ t′ ≤ t}∪{(t′, r′) : t− 1 ≤ t′ ≤ t, r′ ≥ 1}in [16,18], see fig. 2(a);

CCRA` (t) = {(t′, r′) : ` ≤ r′ ≤ 1, t− r′ ≤ t′ ≤ t}

in [19], see fig. 2(b).

(3)

In this contribution, we will focus on stable laws. In-deed, it appears that for stable cascades the ID stochas-tic measure M may be simplified, and expressed using astochastic integral versus a stable noise. Such a formula-tion was proposed by Schmitt and Marsan [15] and itsproperties in the Gaussian case studied in Schmitt [17]:

Definition 2

ελ(t) = λ−σ2/2 exp(

σ

∫ t

t+1−λ

(t + 1− u)−1/2 dB(u))(4)

where λ plays the role of the inverse of a resolution as1/` in definition 1, dB(u) is a Gaussian noise and σ2 is avariance parameter.

Our first aim is to gather the various definitions abovewithin consistent notations and to precisely formulate thelink between approaches using respectively “multiplica-tive cascades” like (2) and “stochastic integrals” like (4).To bring all approaches within the same context, we willmainly use causal definitions.

2.2 Scaling properties

The scaling properties of IDCs in 1 dimension have beenstudied in Refs. [15–19,25,26]. We recall here some basicresults. The measure M , the distribution G, the controlmeasure m and the geometry of the cone of influence C`(x)

1 Note that the large scale in the definition of C`(t) hasbeen arbitrarily set to 1 without loss of generality. Choosing adifferent large scale L would simply reduce to a change of unitst → t · L, ` → r · L.

F. G. Schmitt, P. Chainais: On causal stochastic equations for log-stable multiplicative cascades 3

t

r

0

1

t

r

0

1

t

r

0

1CBM

` (t) CCRA` (t) CSM

` (t)

` ` `

t− 1 t− ` t t− 1 t− ` t t− 1 t− ` t(a) (b) (c)

Fig. 2. Cone C`(t) chosen by (a) Bacry & Muzy [16], (b) Chainais, Riedi & Abry [19]; (c) Schmitt and Marsan in [15] oncethe correspondence is established (see text).

control the scaling structure as well as marginal distribu-tions of the cascade. One major property of IDCs is thefollowing [15,16,19]:

IE[Qq` ] = exp [−ϕ(q) m(C`)] (5)

whereϕ(q) = ρ(q)− qρ(1), ϕ(1) = 0, (6)

for all q for which ρ(q) = − log G(q) is defined.We may also remark that, turning to local averages κr

over a box of size r ≥ `

κr(t) =1r

∫|t′−t|<r/2

Q`(t′) dt′, (7)

one gets:IEκr(t)q ∝ exp [τ(q) m(Cr(t))] (8)

where in general τ(q) = ϕ(q), at least within some lim-ited range of values of q [25,27]. This corresponds to ob-servables and is therefore denoted “dressed cascades” asopposed to “bare cascades” represented by Q` [28,3]. Wedo not consider this question further here, and mainly fo-cus on the formalism describing the construction of barecascades.

2.3 The Normal case

In this section, the distribution G of definition 1 is theNormal law N (µ, σ2) with average value µ and varianceσ2. It was shown in [16,18] that an exact power law scalingis observed if (see figure 2(a)):

dm(t, r) =dt dr

r2,

C`(t) = CBM` (t)

= {(t′, r′) : ` ≤ r′ ≤ 1, t− r′ ≤ t′ ≤ t}∪{(t′, r′) : t− 1 ≤ t′ ≤ t, r′ ≥ 1}.

(9)

Then, the key quantity is the control measure of the coneof influence C`:

m(CBM` (t)) =

∫ 1

`

∫ t

t−r

dt′︸︷︷︸=r

dr′

r2+∫ ∞

1

∫ t

t−1

dt′︸︷︷︸=1

dr′

r2

= log(1/`) + 1. (10)

The essential term that ensures a multifractal behaviouris the log(1/`) term ; the last“+1” term only ensures exactpower law scaling behaviours for scales ranging from ` to1. In [19], only the 1st term was used. Indeed, as ` → 0,the differences due to the choice of the cone of figure 2(a)or (b) asymptotically disappear, except around t ' 1. Thecontribution of the upper part {(t′, r′) : t−1 ≤ t′ ≤ t, r′ ≥1} is minor indeed.

In [16,18] as in [19,20], the presentations of infinitelydivisible cascades are based on multiplicative cascades andtheir generalizations, as in definition 1. In [17], anotherviewpoint is proposed that evokes random walks throughthe use of a stochastic integral as in definition 2. To clarifythe link between these apparently different approaches, letus note that the integrals in (10) were written integratingfirst w.r. to t and then w.r. to r. Inverting this order andcomputing the same integrals, we obtain (see figure 3):

m(C`(t)) =∫ t−`

t−1

∫ ∞

t−t′

dr

r2︸︷︷︸(t−t′)−1

dt′ +∫ t

t−`

∫ ∞

`

dr

r2︸︷︷︸1/`

dt′

=∫ t−`

t−1

(t− t′)−1 dt′ + 1 (11)

This writing can be used to suggest a rewriting of eq. (2)using stochastic integrals. Eq. (11) implicitly gives a de-composition of the stochastic measure M(C`(t)) as a sumof successive elementary terms in time (see figure 3) whichcorresponds to a sum of independent identically distributednormal variables associated to the vertical slices on fig-ure 3(b). This is due to the fact that M is an independentand additive random measure. This way, the use of a 2D


0

1

t

r

t−10

1

t

r

t−1(a) (b)

Fig. 3. (a) Integrating first w.r. to t, and then w.r. to r: the process is described as a multiplicative cascade; (b) Integratingfirst w.r. to r, then w.r. to t: the process is described as the exponential of a stochastic integral.

random measure is replaced by a 1D stochastic integra-tion.

In the Normal case, one has ρ(q) = −µq − σ2 q2

2 in(1). Thus, from (11), the random variable associated toan elementary slice E of width dt′ around t′ has mean∝ µm(E) and variance∝ σ2m(E) where m(E) = dt′/(t−t′)if ` ≤ t− t′ ≤ 1 and m(E) = 1/` if t− t′ ≤ `. As a conse-quence, for some given instant t we get the following newexpression for M(C`(t)), written as a stochastic integral(see also Appendix A):

M(C`(t)) , µ ·m(C`(t)) + σ

∫ t−`

t−1

(t− t′)−1/2 dB(t′)

+ σ

∫ t

t−`

`−1/2 dB(t′). (12)

where , means “equality in distributions” and m(C`) isgiven by (10) when C` = CBM

` . Equivalently, this can canbe written in the form

M(C`(t)) , µ ·m(C`(t)) + σ

∫ t

t−1

K`(t− t′) dB(t′), (13)

with

K`(τ) ={

τ−1/2, ` ≤ τ ≤ 1,

`−1/2, 0 ≤ τ ≤ `,(14)

when C` = CBM` as in (9). Thus, for any instant t, the

random variable Q`(t) associated to an infinitely divisiblecascade as defined by (2) can be described thanks to someBrownian motion dB(t′) by

Q`(t) ,eµm(C`)

e(µ+σ2/2)m(C`)exp

[σ

∫ t

t−1

K`(t− t′) dB(t′)]

, e−σ22 m(C`) exp

[σ

∫ t

t−1

K`(t− t′) dB(t′)]

(15)

where e−σ22 m(C`) is a normalization factor.

On the way back to the formulation (4) proposed bySchmitt and Marsan in [15], let ` = 1/λ. The change ofvariable t′ = t− 1

t+1−u in (4) yields:

ελ(t) = `σ2/2 exp

(σ

∫ t−`

t−1

(t− t′)−1/2 dB(t′)

), (16)

where one recognizes exactly the exponential of the firstterms of (12) with

CSM` (t) = {(t′, r′) : t− 1 ≤ t′ ≤ t− `, r′ ≥ t− t′}. (17)

Then, m(CSM` ) = log(1/`) and the normalization factor

e−σ2/2m(C`) in (15) becomes `σ2/2 in (16). The cone CSM`

is illustrated on fig. 2(c). The difference between this coneand the cone of figure 2(a) is a thin strip{(t′, r′) : t− ` ≤t′ ≤ t, r′ ≥ `} that becomes infinitely thin and negligibleas ` → 0. We have indeed shown that this formulation isclosely related to (2).

Last, note that choosing the cone proposed by Chainaiset al. in [19] –see figure 2(b)– yields

M(CCRA` (t)) , µ·log(1/`)+σ

∫ t−`

t−1

(1

t− t′− 1)1/2

dB(t′)

+ σ

∫ t

t−`

(1`− 1)1/2

dB(t′). (18)

Again, the difference between (12) and (18) is sensitivefor ` ' 1 only, and asymptotically disappears as ` → 0.

It is important to note that the equalities in distri-bution in equations (12) to (15) are valid for one giveninstant t only. Indeed, (15) could not be written simul-taneously with the same Brownian motion for two dis-tinct instants t1 6= t2. Therefore, ελ(t) in (16) and Q`(t)


in (15) are not identical processes. One can simply saythat, for a given instant t, the random variables Q`(t) andελ(t) are equal in distribution. Something has been “lost”from Q`(t) to ελ(t): the independence of the (time,scale)stochastic measure M in scale is no more present in thedefinition of ελ where only the integration over time re-mains. Given a Brownian motion dB(t′), the correlationsof the process ελ(t) are controlled by the kernel K`(τ).However, it was shown in [17] that the essential multi-fractal properties have been preserved. In particular, onerecovers the characteristic property of multiplicative cas-cades cov(log ελ(0), log ελ(τ)) ∼ σ2 log(1/τ). This impor-tant remark remains valid for all the generalizations con-sidered below.

2.4 The stable non-Gaussian case

Previous results generalize easily to the stable case, i.e.,for the choice G = S(α, σ, µ, β) [22], provided the law G isasymmetrical with β = −1, this condition being imposedby the fact that we need some moments of the exponentialof the stable stochastic process to exist (see Schertzer andLovejoy [3,2] and Kida [29]). Then, the function ρ(q) in (1)takes the form ρ(q) = −µq−σαqα with 0 < α < 2 (α 6= 1).

The same framework as before gives eq. (13) with aLevy stable measure dLα with parameter 0 ≤ α ≤ 2.From (11), the random variable associated to an elemen-tary slice E of width dt′ around t′ has now mean ∝ µm(E)and scale parameter∝ σαm(E) where again m(E) = dt′/(t−t′) if ` ≤ t− t′ ≤ 1 and m(E) = 1/` if t− t′ ≤ `. Therefore,this provides the following expression:

M(C`(t)) , µ ·m(C`(t)) + σ

∫ t

t−1

K`(t− t′) dLα(t′), (19)

where the kernel is here:

K`(τ) ={

τ−1/α, ` ≤ τ ≤ 1,

`−1/α, 0 ≤ τ ≤ `.(20)

The choice α = 2 gives again the Gaussian case.

3 The 2-dimensional case

A generalization of infinitely divisible cascades to D di-mensions was proposed in [20]. Examples of realizationsin 2D and 3D generated using the algorithms proposed inthis reference are shown on figure 1. For pedagogical rea-sons, we first consider the 2D case in this section beforegoing to D-dimensional generalization in the next section.For D = 2 dimensions we have for X ∈ R2

dm(X, r) = C(2)dr

r3dX (21)

where C(2) is a constant that will be estimated below,and X = (x, y) is a point in the 2D (x, y) space. The firstsubsection below deals with the 2D spatial case and thesecond subsection deals with the causal case where the ydimension is replaced by time t ≡ y.

r

x

y

Qx

r=1

r=

(x,y)

ρ = ` ρ = 1

A`(x) B`(x)

Fig. 4. Cone C`(x = (x, y)) for D = 2 decomposed in twoparts A`(x) and B`(x).

3.1 The two-dimensional spatial case

In the 2D spatial case, the quantity M(C`) results from anintegral over a conical volume defined by:

C`(X) = {(X′, r′) : ` ≤ ‖X′ −X‖ ≤ 1, r′ ≥ ‖X′ −X‖}∪{(X′, r′) : ‖X′ −X‖ ≤ `, r′ ≥ `}. (22)

as described on figure 4. Here the cone belongs to a 3dimensional space: 2 dimensions for the vector X = (x, y)and 1 dimension for the scale r. We must also underlinethat the norm ‖.‖ in the above equation concerns the 2-dimensional projection of a point having a 3D position(X, r).

The key quantity to compute is m(C`(X)).

m(C`(X)) =∫∫

`≤‖X′−X‖≤1

(∫ ∞

‖X′−X‖

C(2)dr

r3

)dX′

+∫∫

‖X′−X‖≤`

(∫ ∞

`

C(2)dr

r3

)dX′ (23)

where we have again integrated w.r. to r first as in (11). Wethen note that the integration over the conic volume canbe decomposed into a sum of thin cylindrical tubes. Wewill therefore use below cylindrical coordinates with ρ =‖X′−X‖. Let SD the surface of the D dimensional unitarysphere (e.g. 2π in 2 dimensions, 4π in 3 dimensions...).This yields in 2 dimensions:

m(C`(X)) =∫ 1

`

(∫ ∞

ρ

C(2)dr

r3

)S2ρdρ (24)

+∫ `

0

(∫ ∞

`

C(2)dr

r3

)S2ρdρ

=C(2)S2

2

(log(

1`

)+

12

)= log

(1`

)+

12

ifC(2)S2

2= 1. (25)

which does not depend on X. This last condition pre-scribes the choice C(2) = 1/π. Then, self-similarity is en-


sured in the same conditions as in dimension 1, due to thechoice of the control measure in (21).

By computing the integrals w.r. to r in (23), we getfrom arguments similar to those used in 1 dimension (seealso Appendix 2) the following expression for any givenposition X:

M(C`(X)) , µ ·m(C`)

+ σ

∫∫‖X′−X‖≤1

K(2)` (‖X′ −X‖) dLα(X′), (26)

where the new kernel is:

K(2)` (ρ) =

{S−1/α

2 ρ−2/α, for ` ≤ ρ ≤ 1,

S−1/α2 `−2/α, for 0 ≤ ρ ≤ `,

(27)

for 0 < α ≤ 2. For α < 2, dLα is, as before, an α-stablerandom measure with asymmetric parameter β = −1. Sec-tion 4 presents the general version of this result in D di-mensions.

3.2 The 2D space-time causal case: X = (x, t)

It was also proposed in [20] to use multi-dimensional in-finitely divisible cascades to generate time evolving pro-cesses. Here we consider the 2D space-time causal case.The y dimension is now considered as a time variable anddenoted t ≡ y so that X = (x, t). Our purpose is to showthat the previous construction can be used to build sometime evolving scalar field. The resulting field can be seenas a 1D spatial multifractal field obeying some multifractaltime evolution. Causality is guaranteed by performing astochastic integration over a cylinder analogous to the onein previous section but with the further condition t′ ≤ t(or y′ ≤ y), see fig. 5. The previous approach applies withonly some changes due to the introduction of causality. Forinstance, the normalizing constant C(2) becomes 2/π inplace of 1/π. The previous modifications of the stochasticintegration for stable random measures still apply, and weobtain here the following expression for any given positionX:

M(C`(X)) , µ ·m(C`) + σ

∫∫F

K(2)` (‖X′ −X‖) dLα(X′),

(28)where X = (x, t) and F = {X′ = (x′, t′) : ‖X′ − X‖ ≤1; t′ ≤ t}: see Fig 5. The 2D causal kernel is still givenby equation (27). Section 4 presents the general version ofthis result in D dimensions.

Then, the origin of temporal, respectively spatial, cor-relations in the resulting process Q`(x, t) can simply beexplained geometrically. The correlations are governed bythe measure of the intersection of half-hyperspheres in the(x, t) domain. Figure 6(a) illustrates the origin of sharedinformation between successive values of the process atsome given position, Q`(x, t1) and Q`(x, t2). Figure 6(b)shows the origin of spatial correlations between Q`(x1, t)and Q`(x2, t) at some given instant t. Figure 6(c) shows

Fig. 5. Spatial domain of integration corresponding to a stablerandom measure for a 2D stable process on a space (x, t).

the most general situation for different time instants t1 6=t2 and space positions x1 6= x2.

Such a time dependent positive valued process Q`(x, t)exhibits a multifractal behaviour both in time and space.Moreover, it can be considered as more realistic than theapproach proposed in [30–32]. Indeed, in these works, tem-poral correlations at large time intervals are controlled bythe same quantities that control spatial correlations oversmall distances. This is inherent to their construction. Weemphasize that there is not such a paradox in the presentdefinition: temporal correlations over large time intervalsare not dominated by spatial correlations over small dis-tances.

4 The D-dimensional case with D ≥ 2

4.1 The spatial case

As shown in [20], the definitions naturally extends from 1to D > 1 dimensions by using

dm(X, r) = C(D)dr

rD+1dX (29)

where C(D) is a constant linked to the surface of a hyper-sphere of dimension D, and chosen to ensure the adequatescaling law.

The definition of C`(X), the conical volume in D-dimen-sion is still given by equation (22), where the cone belongsto a dimension D + 1: D dimensions for the vector X and1 dimension for the scale r. As before, the norm ‖.‖ con-cerns the D-dimensional projection of a point of position(X, r).

Computations similar to those of (23) & (25) per-formed in D dimensions yield:

m(C`(X)) =∫ 1

`

(∫ ∞

ρ

C(D)dr

rD+1

)SDρD−1dρ

+∫ `

0

(∫ ∞

`

C(D)dr

rD+1

)SDρD−1dρ (30)


x

t

(x,t1) (x,t

2)

x1

x2

t

(x1,t)

(x2,t)

x

x1

x2

t

(x1,t

1)

(x2,t

2)

x

(a) (b) (c)

Fig. 6. Illustration of the origin of (a) temporal, respectively (b) spatial, correlations in the resulting process Q`(x, t). Thecorrelations can be described in terms of the measure of the intersection of half-hyperspheres in the (x, t) domain.

so that C(D) = D/SD = DΓ (D/2)/(2πD/2) since SD =2πD/2/Γ (D/2) where Γ is the gamma function. This choicefinally yields

m(C`(X)) = log(

1`

)+

1D

. (31)

which does not depend on X. Therefore, the previous ap-proach for D = 2 for an α-stable stochastic measure gen-eralizes to D dimensions providing an expression with apolar isotropic kernel:

M(C`(X)) , µ ·m(C`)

+ σ

∫∫‖X′−X‖≤1

K(D)` (‖X′ −X‖) dLα(X′), (32)

where the new kernel is for 0 < α ≤ 2:

K(D)` (a) =

{S−1/α

D ρ−D/α, for ` ≤ ρ ≤ 1,

S−1/αD `−D/α, for 0 ≤ ρ ≤ `.

(33)

For D = 1 and D = 2, we recover the kernel describedrespectively by (20) and (27). As a result, we get a newdescription of the D-dimensional version of Q` as definedby (2) by using a stochastic integral over the unitary disc{X′ : ‖X′ − X‖ ≤ 1} (in D dimension). Interestingly,this can be compared to the heuristic derivation of the D-dimensional stable case presented in [3,2]. These authorsalso introduced a power-law integration kernel over somevolume, but the exponent was not the same: K(ρ) ≈ ρ−γ

with γ = 1−D + D/α.An important remark concerns the simulation of log-

stable multiplicative cascades. They are usually ratherpainful to simulate because they do not belong to thefamily of compound Poisson cascades, see [19]. The resultspresented here and eq. (32) in particular give us a muchmore simple way to simulate such processes. One mustfirst simulate the independent random measure dLα(X),which remains to simulate a set of i.i.d. stable variablesover some regular discrete sampling of the X space. Then,one simply uses eq. (32) to get M(C`(X)). Taking the ex-ponential yields Q`(X).

Once again, it is important to note that equation (32)holds for one given position X only to which dLα(X′) is

indeed associated. Therefore, we emphasize that the pro-cess ελ(X) obtained by taking the exponential of (32) isnot identical to Q`(X) as defined by (2). However, as al-ready mentioned in section 2.3, the essential multifractalproperties are preserved.

4.2 The causal spatio-temporal case

We come back to the use of D-dimensional infinitely divis-ible cascades to build time evolving processes as alreadyevoked in section 3.2 and introduced in [20]. Some “multi-fractal films” can be synthesized this way 2. In this spirit,the same approach as above for 2-dimensional causal cas-cades can be easily generalized to D dimensions by usingX = (x, t). Again, we will use half of a cone C` as definedabove: the half for which t′ ≤ t. The resulting field can beseen as a D−1-dimensional multifractal field, with a timeevolution. The previous simplifications of the random in-tegration for stable random measures still apply, and weobtain the following final expression:

M(C`(X)) , µ ·m(C`) + σ

∫∫F

K(D)` (‖X′ −X‖) dLα(X′),

(34)where X = (x, t) and F = {X′ = (x′, t′) : ‖(x′, t′) −(x, t)‖ ≤ 1; t′ ≤ t}. The D-dim causal kernel is still givenby equation (33).

The comments on figure 6 are still relevant in this gen-eral context. As a consequence, this construction inheritssome nice geometrical understanding. Such a descriptionmakes explicit:

– the stationarity of the process at fixed position xo ;– a causal time evolution ;– a description in terms of a stochastic integral with re-

spect to a stable noise dLα ;– a correspondence between this description where time

and space variables are explicitely distinguished and

2 See http://www.isima.fr/∼chainais/PUB/software.html;an example of a multifractal film, obtained by simulating a2+1 dimensional field and considering 2 spatial dimensionsand 1 time dimension.


the global description expM(C`(x, t)) where x ∈ Rd

and t ∈ R are indeed merged.

Such a construction can be compared to the fraction-ally integrated fields described in [3] and later extendedto causal (D+1) space-time multifractal processes in [34].In the latter, a Fourier approach was proposed to im-pose the fundamental property cov(log ελ(0), log ελ(x)) ∼σ2 log(1/‖x‖) thanks to k−1 filtering in the Fourier do-main. The resulting processes are causal, but, due to theuse of a Fourier transform, they could not be synthesizedin a causal manner. Moreover, such a Fourier approach ismemory demanding. The definitions in terms of stochas-tic integrals proposed here overcome such drawbacks andprovide an explicit description directly in the space-timedomain.

Moreover, note that definition (34) could be formu-lated in the framework of Generalized Scale Invariance [33,35] as well. This framework permits to introduce some con-troled anisotropy by replacing the classical self-similarityproperty in D + 1 dimensions by a matrix operator self-similarity. For instance, this can be achieved by using (seeequation (44) in [34])

‖(x, t)‖ = (|x|del−D/α + |t|del−D/α

1−H )1

del−D/α

where H characterizes the anisotropy (H = 0 in the isotropiccase) and del = D+1−H. This could be helpful to under-stand better how large-scale anisotropy in turbulent flowshave influence at small scales. Indeed, in Kolmogorov’shomogeneous turbulence framework, it is often assumedthat the cascade process “washes” the details of the large-scale flow, so that after few cascade steps, small scalesbecome locally isotropic. In fact experimental results in-dicate that this may not be true, and the present approachmay provide a theoretical framework to develop continu-ous anisotropic and scaling models.

5 Conclusion and discussion

In this paper, we have first given a consistent presentationof various definitions and presentations of stable cascades,a special family of infinitely divisible cascades, in 1 dimen-sion. To this aim, we have used both the multiplicativecascade viewpoint as in [15,16,18–20] and the stochasticintegral viewpoint as in [17]. We have made explicit the dif-ferences between definitions mainly in terms of the chosencone C` when using the multiplicative cascade approach orin terms of the chosen integration domain when using thestochastic integral approach. In the latter, the integrationkernel has been identified.

We have also considered a D dimensional generaliza-tion, showing that the simplification arising in the 1Dcase for stable random measures can also be introducedin D-dimension, using cylindrical coordinates. We haveprovided explicit expressions, using an integration over aD-dimensional space, with a kernel whose general expres-sion was provided. We have first given the 2D case beforegeneralizing straightforwardly to D ≥ 2. In each case we

Fig. 7. Vertical plate denoted T (r0, a, b) in the text, in the(t, r) space.

have also considered the spatio-temporal situation whereX = (x, t), which is built using the same kernel as forthe purely spatial situation; only the integration domainis different, to ensure causality.

A general formulation in terms of a stochastic integralwith respect to a Levy noise allowed us to make clear thefundamental properties of such fields: multifractality bothin time and space, homogeneity, stationarity and causal-ity. The extension to the framework of generalized scaleinvariance as in [34] is considered. Such a time depen-dent positive valued process ε`(x, t) exhibits a multifrac-tal behaviour both in time and space. Moreover, it can beconsidered as more realistic than the approach proposedin [30–32], since in these works, the same quantities controlboth spatial correlations over small distances and tempo-ral correlations over large time intervals. We emphasizethat this is not the case in our definition and that thecorrelation function receives a quite intuitive geometricalinterpretation (see fig. 6).

Finally, an important remark concerns the simulationof log-stable multiplicative cascades. They are usually ra-ther painful to simulate because they do not belong to thefamily of compound Poisson cascades and cannot be ob-tained from a simple marked point Poisson process. Thepresent results provide a much more simple way to thesimulation of very similar α-stable processes. The frame-work presented and developed here corresponds to D-dimensional scalar multifractal processes. For practical ap-plications, this is helpful to simulate scalar processes; how-ever this cannot be directly used for the simulation of realflows, since turbulence is fully tensorial. going to real flowsituations would need to generalize this continuous scalarframework to continuous multifractal tensorial processes.The corresponding theoretical framework is still to be de-veloped. If turbulent cascade models could be adapted tovectorial or tensorial frameworks, it could lead to muchmore reliable predictive models for industrial flows, thanthe eddy-viscosity models, which are usually only able tocorrectly predict moments of order 1 and 2.


A Stability property for random measures onthin strips: the 1D case

This appendix is aimed at giving a hint to the computa-tions used to link the stable stochastic measure of someset to some sotchastic integral with respect to some stablenoise in dimension 1.

We consider here a simple case, illustrated by Figure7. We denote this set for b > a, T (r0, a, b) = {(t′, r′) : a ≤t′ ≤ b; r0 ≤ r′}. We have m(T (r0, a, b)) = (b− a)/r0. Andwe consider here a Gaussian process with ρ(q) = − 1

2q2, sothat:

IE[exp [qM(T (r0, a, b))]] = exp[q2(b− a)/2r0

](35)

Let us note here ω a Gaussian random measure of varianceσ2 and integrated on an interval (a, b):

ω = W (σ, a, b) = σ

∫ b

a

dB(t) (36)

We then have:

IE[exp [qω]] = exp

[q2

2

∫ b

a

σ2dt

]= exp

[q2(b− a)σ2/2

](37)

Comparing equations (35) and (37), we see that the stablerandom variable M (T (r0, a, b)), which is a random mea-sure on a 2D space, is then equal in distribution to a 1Dprocess, following the stability property. This gives thefollowing relation for a Gaussian random measure:

M (T (r0, a, b)) , W (r−1/20 , a, b) (38)

where , means “equality in distribution”. Applying thisrelation to the thin plate T (r0, t, t+dt) gives the following:

M (T (r0, t, t + dt)) , r−1/20 dB(t) (39)

For the stable case of basic index 0 ≤ α ≤ 2, we get:

M (T (r0, t, t + dt)) , r−1/α0 dLα(t) (40)

B Stability property for random measures ona cylindrical corona: the 2D case

We consider here the case of a void cylinder, illustrated byFigure 8. We denote this set for b > a, V C(r0,X0, a, b) ={(X′, r′) : a ≤ ‖X′ −X0‖ ≤ b; r0 ≤ r′}. On one hand, themeasure of this set is given by:

m (V C(r0,X0, a, b)) = C(2)π(b2 − a2)∫ ∞

r0

dr

r3

= πb2 − a2

2r20

C(2). (41)

where C(2) is the normalizing constant in 2 dimensions.

Fig. 8. Void cylinder denoted V C(r0,X0, a, b) in the text, inthe (x, y, r) space.

We consider here a Gaussian process M(V C(r0,X0, a, b))depending on X0 and such that

log IE[exp [qM(V C(r0,X0, a, b))]] = q2πb2 − a2

4r20

C(2)

(42)On the other hand, let us note here ω a Gaussian ran-dom measure of variance σ2C(2) and integrated over acylindrical corona of inner radius a and outer radius b,centered at point X0: ω = σC(2)1/2

∫A

dB(x, y), withA(X0, a, b) = {X′ : a ≤ ‖X′ −X0‖ ≤ b}. We have:

log IE[exp [qω]] = σ2q2πb2 − a2

2C(2) (43)

Comparing equations (42) and (43), we see that the ran-dom variable M (V C(r0,X0, a, b)), which is a random mea-sure on a 3D volume, is then equal in distribution to arandom measure on a 2D surface, with variance σ2 =C(2)/2r2

0. This gives the following relation:

M (V C(r0,X0, a, b)) , (C(2)/2r20)

1/2M (A(X0, a, b))(44)

For the stable case of basic index 0 ≤ α ≤ 2, we have:

M (V C(r0,X0, a, b)) ,(C(2)/2r2

0

)1/αM (A(X0, a, b))

(45)

C Stability property for random measures onhypercylindrical coronas: the D-dimensionalcase

The previous approach for dimension 2 can be general-ized for D dimension. In this case, the only differenceis the introduction of the term r−D−1 instead of r−3 inthe integral of eq. (41). We still consider a void cylinder


V C(r0,X0, a, b) = {(X′, r′) : a ≤ ‖X′ −X0‖ ≤ b; r0 ≤ r′}and provide here directly the stable case of index α:

M (V C(r0,X0, a, b)) ,(C(D)/DrD

0

)1/αM (A(X0, a, b))

(46)

The author gratefully acknowledge Nicolas Perpete for stimu-lating discussions and comments.

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